Neutron Scattering

determination of accurate atomic parameters (positional and thermal parameters, site
occupations) of lighter elements in the presence ofheavy ones .
The contrast in conventional X-ray diffraction is directly related to the ratio ofthe number of
electrons Zj of thé différent atoms or ions j involved . The atomic scattering factor f in thé
structure-factor formula, which represents thé Fourier transform of thé atomic electron
density distribution, is proportional to Zj (f = Zj for sin9/A, = 0) . Standard X-ray techniques
can hardly differentiate between atoms/ions of a similar number of electrons, and only an
average structure - including a total occupation probability of mixed occupied sites - may be
obtained in such cases .
For neutrons thé atomic scattering factor f is replaced by thé nuclear scattering length (or
coherent scattering amplitude) bj, which is ofthé same order of magnitude for all nuclei but
varies from nucleus to nucleus in a non-systematic way . b,, values, which can be either
positive or negative, depend on thé isotopes and nuclear spin states of thé element j . A
nucleus of an isotope with spin I may have two différent neutron scattering lengths : one for
thé combined spin state J = I + %2 and one with J = I - '/z . An important and fundamental
example is provided by thé simplest of all nuclei, thé proton with spin I = 1/2 . The two spin
states, J = 1 (triplet) and J = 0 (singlet), with statistical weights 3/4 and 1/4 respectively, have
thé scattering lengths for afree proton :
b SH=-23 .7 fin, b'H=+5 .38 fin, bfre,H = 1/4b'H + 3/4b'H = -1 .89 lin (with 10 -15 m = 1 lin).
The value for thé bound proton in a ciystal structure, which is to be used in thé structure
factor calculations, amounts to bl, = 2 -bfreeFl = -3 .741 fm.
The natural isotope mixture and a statistical spin-state distribution lead to thé commonly used
general formula bj = with thé sum of thé différent isotope fractions
a+13+y+. . . = 1 (boa, bop, bj,, being thé individual scattering lengths of thé différent isotopes of
thé element j) . The natural nickel isotopes, for instance, have extremely différent coherent
scattering amplitudes :
b( 58Ni) = +14 .4 fin, b(6°Ni) = +3 .0 fin, b(61Ni) = +7 .6 fm,b(62Ni) = -8 .7 fin, b(64Ni) = -0.37 fin
resulting in an overall scattering length bNi = +10 .34 fm .
Neutron experiments frequently make use of compounds containing single isotope elements,
like fully deuterated samples . Incoherent scattering due to a statistical distribution of isotopes
and nuclear spin states is not discussed here . It may influence thé effective absorption and thé
background conditions ofneutron diffraction studies .
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